JP 2014-507779 A (hereinafter referred to as “Document 1”) describes a Light Emitting Diode (LED) lighting system (light source device) as a conventional example. The LED lighting system described in Document 1 includes: an LED module (light source) including a substrate and an LED element (a light-emitting element) attached to the substrate; and a recycling reflector (a reflector) provided in front of the LED module. The recycling reflector has a transmission opening (opening) through which light from the LED module is transmitted. The recycling reflector includes an inner surface (a reflective surface) which is spherical with respect to the center of the LED element. The light emitted from the LED element and arrived at the inner surface without traveling to the transmission opening is reflected in a direction in which the light returns to the LED element itself. Some rays of the light returned to the LED element are reflected off the LED module and are extracted via the transmission opening to the outside of the recycling reflector. In this way, the LED lighting system enables efficient extraction of the light from the LED module to the outside of the recycling reflector.
However, the LED lighting system is not suitable for reflecting light emitted from an area of the substrate which does not correspond to the center of the inner surface of the recycling reflector by the inner surface to return the light to the LED element and extracting the light via the transmission opening to the outside of the recycling reflector. That is, a large portion of the light emitted from the area which does not correspond to the center of the inner surface is lost before the light is extracted to the outside of the recycling reflector, and thus, the light is not efficiently extracted to the outside of the recycling reflector.